Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Which expressions are equivalent to [tex]-9\left(\frac{2}{3} x+1\right)[/tex]? Check all that apply.

A. [tex]-9\left(\frac{2}{3} x\right) + 9(1)[/tex]
B. [tex]-9\left(\frac{2}{3} x\right) - 9(1)[/tex]
C. [tex]-9\left(\frac{2}{3} x\right) + 1[/tex]
D. [tex]-6 x + 1[/tex]
E. [tex]-6 x + 9[/tex]
F. [tex]-6 x - 9[/tex]

Let's solve the problem by applying the distributive property and then comparing the given choices.

The original expression is:
[tex]\[ -9\left(\frac{2}{3} x + 1\right) \][/tex]

To simplify this expression, let's distribute [tex]\(-9\)[/tex] to both terms inside the parenthesis:

1. Distribute [tex]\(-9\)[/tex] to [tex]\(\frac{2}{3}x\)[/tex]:
[tex]\[
-9 \cdot \frac{2}{3}x = -\frac{18}{3}x = -6x
\][/tex]

2. Distribute [tex]\(-9\)[/tex] to [tex]\(1\)[/tex]:
[tex]\[
-9 \cdot 1 = -9
\][/tex]

So, the simplified expression is:
[tex]\[ -6x - 9 \][/tex]

Now, let's evaluate each of the given options to see if they are equivalent to [tex]\(-6x - 9\)[/tex]:

1. [tex]\(-9\left(\frac{2}{3} x\right) + 9(1)\)[/tex]:
[tex]\[ -6x + 9 \][/tex]
This is not equivalent to [tex]\(-6x - 9\)[/tex].

2. [tex]\(-9\left(\frac{2}{3} x\right) - 9(1)\)[/tex]:
[tex]\[ -6x - 9 \][/tex]
This is equivalent to the original expression.

3. [tex]\(-9\left(\frac{2}{3} x\right) + 1\)[/tex]:
[tex]\[ -6x + 1 \][/tex]
This is not equivalent to [tex]\(-6x - 9\)[/tex].

4. [tex]\(-6 x + 1\)[/tex]:
This is not equivalent to [tex]\(-6x - 9\)[/tex].

5. [tex]\(-6 x + 9\)[/tex]:
This is not equivalent to [tex]\(-6x - 9\)[/tex].

6. [tex]\(-6 x - 9\)[/tex]:
This is equivalent to the original expression.

The expressions that are equivalent to [tex]\(-9\left(\frac{2}{3} x + 1\right)\)[/tex] are:

[tex]\[ \boxed{-9\left(\frac{2}{3} x\right) - 9(1) \quad \text{and} \quad -6x - 9} \][/tex]